Intrinsic Local Modes
Consider a chain of particles of mass m where the
nearest-neighbors are connected by the anharmonic springs. The
anharmonic interparticle potential has the following form
V = K2 x2 / 2 +
K4 x4 / 4 ,
are the harmonic and quartic anharmonic terms, respectively, and
x is the deviation of the spring's length from its
Such lattice supports intrinsic local modes (ILMs) with their
frequencies above the phonon band characterized by the maximal
harmonic plane waves frequency
The eigenvector of the intrinsic local mode can be found within the
rotating-wave apporximation (RWA) where the displacement of the
nth particle from its equilibrium position
is described by the following ansatz
is the amplitude of the mode, and
characterizes its ac displacement pattern. Substitution of the above
ansatz into the classical equations of motion
allows one to find the mode eigenvector.
The ILM's eigenvector is a wave package which transfers to a lattice
envelope soliton in a limit of a weak anharmonicicty.
A similar ansatz can give the eigenvector of a moving ILM.
A more complete description of the intrinsic local modes you can find in
a review article:
S. A. Kiselev, S. R. Bickham, and A. J. Sievers,
"Properties of Intrinsic Localized Modes in One-Dimensional Lattices",
Comments Cond. Mat. Phys, 17, 135-173 (1995).
The above applet allows you to
watch vibrating ILMs in the lattice of 15 particles with periodic
boundaries. The evolution of the chain is calculated by the
molecular-dynamics technique. The parameters of the lattice are the
You can launch either an Odd-Parity ILM
(when a central particle has the highest amplitude)
or an Even-Parity ILM
(when two central particles have the highest and opposite amplitudes).
You can also launch a Moving ILM.
The time is shown in units of the shortest period of
small amplitude plane wave vibrations,
Energy is shown in arbitrary units. The kinetic
energy of the particle and the potential
energy of the bond are shown as the red and the
yellow bars, respectively.
If you wait for a while you will see a spectrum of the particles'
vibrations. It will be shown in the left panel.
As the time of the evolution goes the spectrum resolution improves.
The frequency unit is the maximal plane wave frequency,
Last modified: December 1, 1996